arXiv subjects

D. Holcman

Publications and source records attributed to D. Holcman.

29 records · Page 2Linked to original sources

arXiv2007

Toward a quantitative analysis of virus and plasmid trafficking in cells

Intracellular transport of DNA carriers is a fundamental step of gene delivery. We present here a theoretical approach to study generically a single virus or DNA particle trafficking in a cell cytoplasm. Cellular trafficking has been studied experimentally mostly at the macroscopic level, but very little has been done so far at the microscopic level. We present here a physical model to account for certain aspects of cellular organization, starting with the observation that a viral particle trajectory consists of epochs of pure diffusion and epochs of active transport along microtubules. We define a general degradation rate to describe the limitations of the delivery of plasmid or viral particles to the nucleus imposed by various types of direct and indirect hydrolysis activity inside the cytoplasm. Following a homogenization procedure, which consists of replacing the switching dynamics by a single steady state stochastic description, not only can we study the spatio-temporal dynamics of moving objects in the cytosol, but also estimate the probability and the mean time to go from the cell membrane to a nuclear pore. Computational simulations confirm that our model can be used to analyze and interpret viral trajectories and estimate quantitatively the success of nuclear delivery.

q-bio.QM↗

arXiv2007

Partially Reflected Diffusion

The radiation (reaction, Robin) boundary condition for the continuum diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary, however, the relation between the reaction (radiation) constant in the Robin boundary condition and the reflection probability is still unclear. In this paper we clarify the issue by finding the relation between the reaction (radiation) constant and the absorption probability of the diffusing trajectories at the boundary. We analyze the Euler scheme for the underlying Itô dynamics, which is assumed to have variable drift and diffusion tensor, with partial reflection at the boundary. Trajectories that cross the boundary are terminated with a given probability and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding Wiener path integral to resolve the non-uniform convergence of the probability density function of the numerical scheme to the solution of the Fokker-Planck equation with the Robin boundary condition, as the time step is decreased. We show that the Robin boundary condition is recovered in the limit iff trajectories are reflected in the co-normal direction. We find the relation of the reactive constant to the termination probability. We show the effect of using the new relation in numerical simulations.

math-ph↗

arXiv2005

Concentration of the first eigenfunction for a second order elliptic operator

We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant $ε$ goes to zero. We assume that the first order term is given by a vector field $b$, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis.

math-ph↗

arXiv2005

The survival probability of diffusion with killing

We present a general framework to study the effect of killing sources on moving particles, trafficking inside biological cells. We are merely concerned with the case of spine-dendrite communication, where the number of calcium ions, modeled as random particles is regulated across the spine microstructure by pumps, which play the killing role. In particular, we study here the survival probability of ions in such environment and we present a general theory to compute the ratio of the number of absorbed particles at specific location to the number of killed particles during their sojourn inside a domain. In the case of a dendritic spine, the ratio is computed in terms of the survival probability of a stochastic trajectory in a one dimensional approximation. We show that the ratio depends on the distribution of killing sources. The biological conclusion follows: changing the position of the pumps is enough to regulate the calcium ions and thus the spine-dendrite communication.

math-ph↗

arXiv2005

The Boundary between Compact and Noncompact Complete Riemann Manifolds

In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultimate, sharp result.

math.DG↗

arXiv2004

Stochastic Chemical Reactions in Micro-domains

Traditional chemical kinetics may be inappropriate to describe chemical reactions in micro-domains involving only a small number of substrate and reactant molecules. Starting with the stochastic dynamics of the molecules, we derive a master-diffusion equation for the joint probability density of a mobile reactant and the number of bound substrate in a confined domain. We use the equation to calculate the fluctuations in the number of bound substrate molecules as a function of initial reactant distribution. A second model is presented based on a Markov description of the binding and unbinding and on the mean first passage time of a molecule to a small portion of the boundary. These models can be used for the description of noise due to gating of ionic channels by random binding and unbinding of ligands in biological sensor cells, such as olfactory cilia, photo-receptors, hair cells in the cochlea.

math-ph↗

arXiv2004

Narrow Escape, Part I

A Brownian particle with diffusion coefficient $D$ is confined to a bounded domain of volume $V$ in $\rR^3$ by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis $a\ll V^{1/3}$ and show that the mean escape time is $Eτ\sim\ds{\frac{V}{2πDa}} K(e)$, where $e$ is the eccentricity of the ellipse; and $K(\cdot)$ is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula $Eτ\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $Eτ=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R}) ]$. This problem is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function.

math-ph↗

arXiv2004

Narrow Escape, Part II: The circular disk

We consider Brownian motion in a circular disk $Ω$, whose boundary $\pΩ$ is reflecting, except for a small arc, $\pΩ_a$, which is absorbing. As $ε=|\partial Ω_a|/|\partial Ω|$ decreases to zero the mean time to absorption in $\pΩ_a$, denoted $Eτ$, becomes infinite. The narrow escape problem is to find an asymptotic expansion of $Eτ$ for $ε\ll1$. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general smooth domain, $Eτ= \ds{\frac{|Ω|}{Dπ}}[\log\ds{\frac{1}ε}+O(1)],$ ($D$ is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is $E[τ| \x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}ε} + \log 2 +\ds{1/4} + O(ε)]$. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}ε}$ is not necessarily large, even when $ε$ is small. We also find the singular behavior of the probability flux profile into $\pΩ_a$ at the endpoints of $\pΩ_a$, and find the value of the flux near the center of the window.

math-ph↗

arXiv2004

Narrow Escape, Part III: Riemann surfaces and non-smooth domains

We consider Brownian motion in a bounded domain $Ω$ on a two-dimensional Riemannian manifold $(Σ,g)$. We assume that the boundary $\pΩ$ is smooth and reflects the trajectories, except for a small absorbing arc $\pΩ_a\subset\pΩ$. As $\pΩ_a$ is shrunk to zero the expected time to absorption in $\pΩ_a$ becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted $Eτ$, as $ε=|\partial Ω_a|_g/|\partial Ω|_g\to0$. We derive a leading order asymptotic approximation $Eτ= \ds{\frac{|Ω|_g}{Dπ}}[\log\ds{\frac{1}ε}+O(1)]$. The order 1 term can be evaluated for simply connected domains on a sphere by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}ε}$ is not necessarily large, even when $ε$ is small. If the absorbing window is located at a corner of angle $α$, then $Eτ= \ds{\frac{|Ω|_g}{Dα}}[\log\ds{\frac{1}ε}+O(1)],$ if near a cusp, then $Eτ$ grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is $Eτ= \ds{\frac{|Ω|}{(d^{-1}-1)D}}(\frac{1}ε + O(1))$.

math-ph↗

arXiv2003

Perturbation Methods and First Order Partial Differential Equations

In this paper, we give explicit estimates that insure the existence of solutions for first order partial differential operators on compact manifolds, using a viscosity method. In the linear case, an explicit integral formula can be found, using the characteristics curves. The solution is given explicitly on the critical points and the limit cycles of the vector field of the first order term of the operator. In the nonlinear case, a generalization of the Weitzenbock formula provides pointwise estimates that insure the existence of a solution, but the uniqueness question is left open. Nevertheless we prove that uniqueness is stable under a C^{1} perturbation. Finally, we give some examples where the solution fails to exist globally, justifying the need to impose conditions that warrant global existence. The last result reveals that the zero order term in the first order operator is necessary to obtain generically bounded solutions.

math-ph↗

arXiv2000

Singular perturbations and first order PDE on manifolds

In this note we present some results concerning the concentration of sequences of first eigenfunctions on the limit sets of a Morse-Smale dynamical system on a compact Riemanniann manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure $μ$ concentrated on the hyperbolic sets of the field. The set of all possible measure turns out to be a sum of a finite Dirac distributions localized at the critical point of the field and absolutely continuous measure with respect to the Lebesgue measure on each limit cycles : the coefficients which appear in the limit measure can be characterized using the concentration theory. In the second part, certain aspects of some first order PDE on manifolds are studied. We study the limit of a sequence solutions of a second order PDE, when a parameter of viscosity tends to zero. Under some explicit assumptions on some vector fields, bounded and differentiable solutions are obtained. We exibit the role played by the limit sets of the dynamical systems and provide in some cases an explicit representation formula.

math.AP↗